Which function has a horizontal asymptote of $y=3$?
A horizontal asymptote for a function \(f(x)\) is a line \(y=k\) such that as \(x \to \infty\) or \(x \to -\infty\), the function values \(f(x)\) approach \(k\). That is, \(\lim_{x \to \infty} f(x) = k\) or \(\lim_{x \to -\infty} f(x) = k\). We are looking for the function that approaches \(y=3\). This typically occurs in functions involving an exponential term that decays to zero, such as \(f(x) = a(b^x) + c\), where the horizontal asymptote is \(y=c\). We examine the behavior of the given functions as \(x\) approaches infinity or negative infinity.
Let's evaluate the options.
For a function of the form \(f(x) = a(b^x) + c\), if \(0 < b < 1\), then as \(x \to \infty\), \(b^x \to 0\), so \(f(x) \to c\). If \(b > 1\), then as \(x \to -\infty\), \(b^x \to 0\), so \(f(x) \to c\). In both cases, the horizontal asymptote is \(y=c\).
Looking at the structure of the options:
Therefore, the function with a horizontal asymptote of \(y=3\) is \(f(x) = 2(4^x) + 3\).
The function \(f(x) = 2(4^x) + 3\) clearly approaches \(3\) as \(x \to -\infty\). Since \(4^x\) becomes arbitrarily small as \(x\) decreases, the term \(2(4^x)\) approaches \(0\). Adding \(3\) to this value results in the function approaching \(3\). Thus, the correct function is \(\boxed{f(x) = 2(4^x) + 3}\).
Understanding horizontal asymptotes is crucial in modeling real-world growth or decay processes that level off over time. For example, if you are studying the temperature of a hot cup of coffee placed in a room, the cooling curve can be modeled by a function that has a horizontal asymptote corresponding to the room temperature. The coffee cools down, but it never drops below the ambient temperature of the room. By identifying the horizontal asymptote, you can predict the long-term state of such systems.